## 1. Introduction

Spatial scales of turbulence in the atmospheric boundary layer have been characterized for a range of conditions (Kaimal et al. 1972), but observations of turbulence scales in oceanic boundary layers have been limited to unstratified tidal flows (Soulsby 1977; Gross and Nowell 1983). Spatial scales of weak turbulence in a strong mean flow can be estimated from time series measurements at a single point by means of the frozen-turbulence hypothesis (Taylor 1938). In many oceanic environments, surface waves complicate the interpretation of point measurements (Lumley and Terray 1983), and determination of turbulence scales requires a spatial array of sensors.

*D*

_{υw}(Δ

*y*), defined by

**x**and (

*x,*

*y,*

*z*) denote the position vector, with

*x*positive onshore,

*y*alongshore, and

*z*positive upward and equal to zero at the bottom; Δ

*y*is the alongshore separation;

**j**is a unit vector in the

*y*direction;

*t*is time; (

*U,*

*V,*

*W*) is the velocity vector; (

*u,*

*υ,*

*w*) is the departure of the velocity vector from its hour-averaged value; and an overbar denotes an hour average.

The analysis is based on the assumption that velocity fluctuations with timescales of less than 1 h are caused by superposed surface waves and turbulence. At the scales resolved by the array, the operation of subtraction in (1) largely removes effects of waves (Trowbridge 1998). In principle, *D*_{υw} (Δ*y*) increases from zero to 2*υ*′*w*′*y* increases from zero to values greater than the correlation scale of the turbulence. Thus, *D*_{υw} (Δ*y*) describes the alongshore scales of the turbulence contributing to the Reynolds shear stress −*ρ*_{0}*υ*′*w*′*ρ* is the density and the subscript 0 denotes a fixed reference value. This component of the Reynolds stress is a dominant term in the alongshore momentum balance in shallow water (Longuet-Higgins 1970; Feddersen et al. 1998; Lentz et al. 1999).

*z,*

*ρ*

_{0}, the vertical buoyancy flux

*B*= (

*g*/

*ρ*

_{0})

*ρ*′

*w*′

*g*is gravity, and the vertical flux of streamwise momentum, here approximated by

*ρ*

_{0}

*υ*′

*w*′

*U*

*V*

*u*′

*w*′

*υ*′

*w*′

*z*/

*L,*where

*L*= |

*υ*′

*w*′

^{3/2}/(

*κB*) is the Monin–Obukhov length, with the empirical von Kármán constant

*κ*≃ 0.40. Monin–Obukhov scaling implies that

*L*

_{c}is a length characterizing the alongshore correlation scale of the stress-carrying turbulence,

*ϕ*is a function that can be derived from semiempirical results obtained in the atmospheric surface layer, and

*N*

^{2}= −(

*g*/

*ρ*

_{0})∂

*ρ*

*z*is the squared buoyancy frequency, which can be positive or negative. Here, oceanic measurements of

*D*

_{υw}and

*z*are fit to a model to determine

*υ*′

*w*′

*L*

_{c}, and these estimates are combined with measurements of

*N*

^{2}to test (2).

## 2. Methods

### a. Measurements

Four acoustic Doppler velocimeters (ADVs) were deployed in an alongshore array on a nearly planar sand beach, with a slope of approximately 0.03, about 60 m north of the pier at the Scripps Institution of Oceanography in La Jolla, California (Fig. 1). The ADVs were mounted on a low-profile, bottom-mounted frame, in an upward-looking orientation, with sample volumes about 0.75 m above the bottom. The ADV array permits estimates of *D*_{υw} at six nonzero equally spaced separations ranging from Δ*y* = 0.5 m to Δ*y* = 3.0 m. A pressure sensor was buried 0.3 m beneath the seafloor approximately 3 m from the center of the ADV array. A pair of sonar altimeters and a pair of temperature sensors were deployed approximately 10 m from the center of the ADV array (Fig. 1). The temperature sensors were 0.10 m above the ADV sample volumes. Ten-minute time-lapse video images obtained hourly (Holman et al. 1993) allowed the extent of the surf zone to be determined during daylight hours.

Sequential 1-h time series sampled at 8 Hz were obtained from each sensor between 19 October 2000 (yearday 295) and 7 January 2001 (yearday 369), shortly before the ADVs were buried by sand on 14 January 2001. Observations by divers indicate that the ADVs were occasionally fouled by kelp, and comparison of the hour-averaged temperature records indicates that the temperature sensors occasionally drifted. Hour-long records from a particular ADV were identified as having been fouled by kelp if the mean or standard deviation of any velocity component from that sensor differed by more than 0.01 m s^{−1} from the corresponding array average. The results presented here are based on 854 1-h records for which all four ADVs were above sand level, none of the ADVs was fouled by kelp, and the two hour-averaged temperature measurements were within 0.1°C of each other. Video images indicate that the array was seaward of the surf zone during these records.

### b. Models of D_{υw}, L_{c}/z, and N^{2}

*D*

_{υw}follows from a semiempirical expression for the cospectrum of streamwise and vertical velocities in the atmospheric surface layer, given by

*y*axis is aligned with the mean velocity; Co

_{υw}is the cospectrum of

*υ*and

*w,*defined so that

^{+∞}

_{−∞}

_{υw}(

*k*

_{y})

*dk*

_{y}=

*υ*′

*w*′

*k*

_{y}is the

*y*component of the radian wavenumber; and

*L*

_{c}is given by

_{υw}(

*k*

_{y}) is

*E*

_{υw}(Δ

*y*), defined to be the even part of

*υ*(

**x**,

*t*)

*w*(

**x**+

**j**Δ

*y,*

*t*)

*D*

_{υw}(Δ

*y*) = 2[

*E*

_{υw}(0) −

*E*

_{υw}(Δ

*y*)]. It follows that (3) is equivalent to

*ξ*is a dummy variable of integration. Equations (3) and (4) are similar to a model proposed by Kaimal et al. (1972) for neutral and stable stratification (

*B*≥ 0), and they are consistent with theory and measurements at large

*k*

_{y}for both

*B*< 0 and

*B*≥ 0 (Kaimal et al. 1972; Wyngaard and Cote 1972). The integral on the right side of (5) depends only on the ratio of Δ

*y*/

*z*to

*L*

_{c}/

*z*and can be evaluated numerically. According to (5),

*D*

_{υw}increases monotonically with Δ

*y*and is within 10% of 2

*υ*′

*w*′

*y*≥ 3

*L*

_{c}.

*ϕ*in (2).

### c. Analysis

Structure functions were computed for each record at each of the six nonzero spatial separations by removing the hour average of alongshore and vertical velocity, computing products according to (1), and averaging over the 1-h record. Estimates of *z* and the mean water depth *h* were obtained from the altimeter and pressure measurements. Estimates of wave statistics were obtained from the pressure and velocity measurements using linear wave theory.

*T*

*z,*where

*T*is the temperature, were obtained by regressing array-averaged estimates of ∂

*T*/∂

*t,*obtained from centered differences of the temperature time series, against array-averaged measurements of −

*w.*This procedure is based on a linearized model of temperature fluctuations given by

*w*and the covariance of ∂

*T*/∂

*t*and

*w*were dominated by surface wave fluctuations at frequencies between approximately 0.05 and 0.5 Hz. The standard deviation of

*w*was much larger than the bottom slope times the standard deviation of

*u,*which suggests that estimates of ∂

*T*

*z*were not affected significantly by terms neglected in (7), such as

*u*∂

*T*/∂

*x,*or by uncertainties in the orientation of the ADVs. The squared buoyancy frequency was estimated from ∂

*T*

*z*by assuming a linear relationship between temperature and density, with a slope of 0.24 kg (m

^{3}°C)

^{−1}, determined from previous temperature and salinity observations near the measurement site (Lerczak 2000).

Estimates of *υ*′*w*′*L*_{c}/*z* were obtained by fitting measurements of *D*_{υw} to (5), with the measured *z*/Δ*y,* using iterative least squares regression. First, the 854 × 6 measurements of *D*_{υw} were regressed against (5) to produce 854 preliminary estimates of *υ*′*w*′*L*_{c}/*z,* representative of the entire measurement period. The measured *z* and *N*^{2} and the preliminary estimates of *υ*′*w*′*κ*^{2}*N*^{2}*z*^{2}/|*υ*′*w*′*υ*′*w*′*κ*^{2}*N*^{2}*z*^{2}/*υ*′*w*′*κ*^{2}*N*^{2}*z*^{2}/|*υ*′*w*′*D*_{υw} within each group were then regressed against (5) to produce 200 final estimates of *υ*′*w*′*L*_{c}/*z* for that group. The preliminary and final estimates of *υ*′*w*′*L*_{c}/*z,* ordered by *κ*^{2}*N*^{2}*z*^{2}/|*υ*′*w*′*υ*′*w*′*L*_{c}/*z* were obtained by a standard procedure based on Gaussian statistics and linearization of the nonlinear regression about the best-fit values of the parameters (e.g., Draper and Smith 1966). In the calculation of error bounds, the six measurements of *D*_{υw} obtained during each 1-h record were assumed to have only 2 degrees of freedom, because concurrent measurements of *D*_{υw} at different spatial separations were not statistically independent (e.g., Cressie 1993).

## 3. Results

During the 70-day measurement period, the significant wave height ranged from approximately 0.5 to 1.5 m, the standard deviation of the cross-shore velocity ranged from approximately 0.2 to 0.5 m s^{−1}, and the hour-averaged alongshore velocity usually was less than 0.1 m s^{−1}, with maximum values of about 0.2 m s^{−1} (Fig. 2). Hour-averaged cross-shore velocities were much smaller than hour-averaged alongshore velocities. Wave incidence angles were within 10° of shore normal. The mean and standard deviation of ∂*T**z* were −0.01° and 0.03°C, respectively. Stable stratification usually occurred during southward alongshore flows (*V**V**z* was larger than the thickness of the oscillatory boundary layer produced by surface waves, estimated at a few centimeters (Grant and Madsen 1979) but was small in comparison with the hour-averaged water depth *h* (0.15 < *z*/*h* < 0.30). Approximately 60% of the variance of the alongshore current was at frequencies below 2 cycles per day, approximately 20% was at semidiurnal frequencies, and the remaining 20% was at frequencies greater than semidiurnal.

Visual observations obtained nearly daily by divers suggest that the seafloor was smooth, with no signs of wave-orbital or larger-scale ripples or bedforms. The seafloor elevation varied by centimeters on timescales of hours, but the differences between the two altimeters indicate there were no bedforms with horizontal scales smaller than the altimeter spacing and vertical scales larger than the altimeter accuracy of about 0.01 m (Gallagher et al. 1998).

The qualitative dependence of the measured structure functions on spatial separation and stratification is consistent with expectations (Fig. 3). The magnitude of *D*_{υw} is smaller than 2*υ*′*w*′*y,* and it asymptotically approaches 2*υ*′*w*′*y* increases. The approach of *D*_{υw} to 2*υ*′*w*′*y* is more rapid for positive *υ*′*w*′*V**υ*′*w*′*V*

*υ*′

*w*′

*c*

_{d}= (1.3 ± 0.1) × 10

^{−3}at 95% confidence (Fig. 4), which is similar to previous estimates of the drag coefficient in nearshore flows under unbroken waves (Feddersen et al. 1998; Trowbridge and Elgar 2001).

The dependence of *L*_{c}/*z* on *κ*^{2}*N*^{2}*z*^{2}/|*υ*′*w*′*κ*^{2}*N*^{2}*z*^{2}/|*υ*′*w*′

## 4. Discussion

The observations indicate that turbulence length scales in the shallow coastal ocean, at heights above the bottom that are small in comparison with the water depth but are larger than the thickness of the wave boundary layer, are influenced by stratification in a manner consistent with local application of Monin–Obukhov scaling (Fig. 5). This result implies that the dynamics of near-boundary turbulence in the shallow coastal ocean and the atmosphere are similar, even though the environments differ. In the atmospheric surface layer, both *υ*′*w*′*B* are nearly independent of *z.* At water depths of a few meters seaward of the surf zone in the ocean, scaling and measurements (Feddersen et al. 1998; Lentz et al. 1999) indicate that Coriolis acceleration is negligible in the hour-averaged alongshore momentum balance and that the dominant terms in the depth-integrated balance are wind stress and bottom stress, implying that *υ*′*w*′*z,* as in the atmospheric surface layer. However, the seafloor usually is idealized as nonconducting for both heat and salt, so that *B* → 0 as *z* → 0, and thus *B* must vary with *z* if *B* ≠ 0. The present results indicate that the dependence of *B* on *z* does not have a dominant effect on the local dynamics of the turbulence.

*β*is an empirical constant, approximately equal to 6.0 (Hogstrom 1988). For the observations used here,

*z*/

*L*was as large as 0.3, which corresponds in (9) to a reduction in eddy viscosity and an increase in ∂

*V*

*z*by a factor of 2.5. Measurements of

*V*

*z*) often are fit to (9) to infer

*υ*′

*w*′

*υ*′

*w*′

*B*→ 0 as

*z*→ 0, is

*υ*′

*w*′

*B*=

*z*∂

*B*/∂

*z,*where ∂

*B*/∂

*z*= constant. For this case, integration of (9) with respect to

*z*gives

*z*

_{0}is the bottom roughness length, which includes the effect of the bottom boundary layer produced by surface waves (Grant and Madsen 1979). For the data discussed here, the second term in square brackets in (10) is substantially smaller than the first, so that stratification has a small effect on the local relationship between the hour-averaged velocity and the turbulent Reynolds shear stress, even though it has an important effect on the vertical gradient of the hour-averaged velocity in (9).

Divergence from Monin–Obukhov scaling at small *k*_{y} is observed in atmospheric measurements during unstable conditions (*B* < 0) and is attributed to large-scale, convective circulations driven by the unstable density structure (Kaimal et al. 1972), which presumably depend on scales in addition to *B,* *υ*′*w*′*z.* Here, the manifestation of similar processes would be *L*_{c}/*z* larger than predicted by (4) and (6) at large, negative *κ*^{2}*N*^{2}*z*^{2}/|*υ*′*w*′

The alongshore turbulence scale during unstable stratification was the same order of magnitude as the water depth *h* (Figs. 3 and 5), but *h* is unlikely to have placed an important constraint on turbulence scale at the measurement height. The quantity *z*/*h,* a dimensionless measure of the importance of *h,* was small, and the measurements indicate no systematic dependence of *L*_{c}/*z* on *z*/*h.* This result is consistent with laboratory measurements in open channels indicating that the water depth affects the turbulence scales only for *z*/*h* > 0.5 (Nezu and Nakagawa 1993). Also, much of the energy appearing at long spatial scales in turbulence statistics referenced to a single coordinate direction, such as *D*_{υw}(Δ*y*) and Co_{υw}(*k*_{y}), is aliased from shorter scales, because wavelengths of individual Fourier components appear longer if the wavenumber is not aligned with the coordinate direction of interest. Thus, the actual scale of the stress-carrying turbulence was smaller than 3*L*_{c}, the value of Δ*y* for which *D*_{υw} is within 10% of 2*υ*′*w*′

The wave-induced strain rate (of order *kσ*_{u}, where *k* is the wavenumber of the surface waves and *σ*_{u} is the standard deviation of *u*), was larger than the strain rates of the hour-averaged velocity field and the stress-carrying turbulence (both of order |*υ*′*w*′^{1/2}/*z*), but straining by the wave-induced velocity field is unlikely to have had an important effect on the turbulence at the measurement position. The measurements indicate no systematic dependence of *L*_{c}/*z* on *kσ*_{u}*z*/|*υ*′*w*′^{1/2}, the ratio of wave-induced to turbulent strain rates. The wave-induced strain rate is oscillatory and therefore is unlikely to have had an important effect when averaged over many wave periods. Also, an order-of-magnitude argument adapted from Phillips (1961) indicates that wave-induced straining of turbulence had a negligible effect on the hour-averaged budget for the turbulent vorticity variance. Although it does not directly address stress-carrying scales, the argument of Phillips (1961) suggests that wave-induced straining had a minor effect on the turbulence.

## 5. Conclusions

Measurements from an alongshore array of near-bottom velocity sensors located approximately 0.8 m above the seafloor in 4-m water depth on an ocean beach were used to estimate the alongshore scales of the turbulent Reynolds stress just seaward of the surf zone. The analysis is based on a spatial structure function that rejects contributions from waves. Estimates of Reynolds stress are consistent with a quadratic drag law, with a drag coefficient similar to previous observational estimates. Turbulent length scales were approximately 2 m during unstable stratification and were reduced by a factor of roughly 2 during stable stratification, in a manner consistent with atmospheric results based on Monin–Obukhov scaling. Effects of finite water depth and straining of turbulence by the wave-induced velocity field were small. The results imply that turbulence in the shallow coastal ocean, just above the oscillatory bottom boundary layer produced by surface waves, is dynamically similar to turbulence in the atmospheric surface layer.

## Acknowledgments

Support was provided by the Office of Naval Research (Marine Geology and Geophysics and Ocean Modeling and Prediction programs) and the National Science Foundation. Britt Raubenheimer helped to design, deploy, and maintain the sensor arrays, contributed significantly to the data analysis, and provided valuable comments. Staff from the Center for Coastal Studies at the Scripps Institution of Oceanography deployed the arrays and kept them operational despite nearly continuous attacks by kelp in cold water with large waves. Janet Fredericks did the preliminary data processing. Steve Lentz and two anonymous reviewers provided insightful comments on an early draft.

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Time series of (a) significant wave height *H*_{s}, (b) std dev *σ*_{u} of the cross-shore velocity, and (c) alongshore velocity *V*

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1122:SSOSNT>2.0.CO;2

Time series of (a) significant wave height *H*_{s}, (b) std dev *σ*_{u} of the cross-shore velocity, and (c) alongshore velocity *V*

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1122:SSOSNT>2.0.CO;2

Time series of (a) significant wave height *H*_{s}, (b) std dev *σ*_{u} of the cross-shore velocity, and (c) alongshore velocity *V*

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1122:SSOSNT>2.0.CO;2

Structure function *D*_{υw} for different spatial separations vs 2*υ*′*w*′^{−4} m^{2} s^{−2}

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1122:SSOSNT>2.0.CO;2

Structure function *D*_{υw} for different spatial separations vs 2*υ*′*w*′^{−4} m^{2} s^{−2}

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1122:SSOSNT>2.0.CO;2

Structure function *D*_{υw} for different spatial separations vs 2*υ*′*w*′^{−4} m^{2} s^{−2}

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1122:SSOSNT>2.0.CO;2

Estimates of −*υ*′*w*′*U*^{2} + *V*^{2})^{1/2}*V**υ*′*w*′*U*^{2} + *V*^{2})^{1/2}*V*^{−3} at 95% confidence. It is not known why the observed relationship between −*υ*′*w*′*U*^{2} + *V*^{2})^{1/2}*V*

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1122:SSOSNT>2.0.CO;2

Estimates of −*υ*′*w*′*U*^{2} + *V*^{2})^{1/2}*V**υ*′*w*′*U*^{2} + *V*^{2})^{1/2}*V*^{−3} at 95% confidence. It is not known why the observed relationship between −*υ*′*w*′*U*^{2} + *V*^{2})^{1/2}*V*

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1122:SSOSNT>2.0.CO;2

Estimates of −*υ*′*w*′*U*^{2} + *V*^{2})^{1/2}*V**υ*′*w*′*U*^{2} + *V*^{2})^{1/2}*V*^{−3} at 95% confidence. It is not known why the observed relationship between −*υ*′*w*′*U*^{2} + *V*^{2})^{1/2}*V*

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1122:SSOSNT>2.0.CO;2

Dimensionless length-scale parameter *L*_{c}/*z* vs dimensionless stratification parameter *κ*^{2}*N*^{2}*z*^{2}/|*υ*′*w*′*L*_{c}/*z.* The vertical error bars represent approximate 95% confidence limits, estimated as explained in the text. The measurements indicate a reduction in length scale with increasing stratification parameter, in agreement with the atmospheric results. The stratification parameter changes sign because *N*^{2} can be positive or negative

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1122:SSOSNT>2.0.CO;2

Dimensionless length-scale parameter *L*_{c}/*z* vs dimensionless stratification parameter *κ*^{2}*N*^{2}*z*^{2}/|*υ*′*w*′*L*_{c}/*z.* The vertical error bars represent approximate 95% confidence limits, estimated as explained in the text. The measurements indicate a reduction in length scale with increasing stratification parameter, in agreement with the atmospheric results. The stratification parameter changes sign because *N*^{2} can be positive or negative

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1122:SSOSNT>2.0.CO;2

Dimensionless length-scale parameter *L*_{c}/*z* vs dimensionless stratification parameter *κ*^{2}*N*^{2}*z*^{2}/|*υ*′*w*′*L*_{c}/*z.* The vertical error bars represent approximate 95% confidence limits, estimated as explained in the text. The measurements indicate a reduction in length scale with increasing stratification parameter, in agreement with the atmospheric results. The stratification parameter changes sign because *N*^{2} can be positive or negative

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1122:SSOSNT>2.0.CO;2

^{*}

Woods Hole Oceanographic Institution Contribution Number 10558.